1.

Introduction

Analyzing light scattering patterns of biological samples is a field of active research with growing interest. The observation of the scattering signatures enables the characterization of structure, size, and refractive index of a sample. For the examination of single cells, a microscopic setup is crucial. Light scattering microscopy allows one to investigate particles not only spectrally resolved,^1,2 but also angular resolved.^3,4 In the literature, several other methods are shown to acquire the scattering signatures in spectral,⁵ angular,^6,7 spatial,⁸ or temporal resolution modes,⁹ for instance. Multimodality microscopic systems combine angular and spectrally resolved measurements.^10,11 Scattering microscopy is a label-free technique which is essential for investigating biological cells.¹² The microscopy setups offer various opportunities of sample illumination. Therefore, miscellaneous setups have been reported with brightfield illumination,¹⁰ darkfield illumination,¹¹ and confocal illumination.^13,14

A novel light scattering microscopy setup with a combination of spectral and angular resolution modes is introduced in this publication. Cottrell et al.¹⁰ and Smith and Berger¹¹ published similar setups. In contrast to those, the microscope shown here has several crucial advantages. The presented microscope has a unidirectional illumination, similar to angular resolved low coherence interferometry systems,^7,15 instead of a rotationally symmetric epi- or transillumination, respectively. Hence, the angular distribution of the scattered light is not integrated over the azimuthal angle and no information is lost. In addition, the orientation of the sample can be rotated with respect to the direction of illumination. These two facts are important for the characterization of nonspherical particles that scatter into a preferred direction, such as spheroidal or cylindrical samples for instance.^15,16 Furthermore, the setup shown here is a microscope with a reflected darkfield illumination, which is advantageous for the investigation of thicker samples or in situ measurements.^7,17 Considering Mie theory, the backward scattered light contains more information than the forward scattered light in both angular and spectral resolution mode.¹⁸ Besides, the system presented in this contribution is based merely on elastic scattering, which is in contrast to the setup of Smith and Berger that combines Raman scattering and elastic scattering.¹¹

For the understanding of the interaction of light with biological cells or tissue, the inner structures of cells can be assessed as a first approximation as spheres,^19,20 cylinders,²¹ or a mixture of spheres and cylinders.²² The scattering of light by spheres²³ and by cylinders²⁴ can be described analytically. Prior to investigating biological cells and tissues, the scattering microscopy setup has to be evaluated by reference samples. The scattering microscope shown here was successfully evaluated in the spectral mode by comparison with a collimated transmission setup by Schmitz et al.²⁵ In this contribution, the angular resolution mode is compared with the spectral resolution mode. For validation, single polystyrene beads are measured angular and spectrally resolved and the sphere diameters are determined. The conformity of angular and spectral resolution mode is verified by comparing the diameter values for each single measurement. Both methods are in a very good agreement.

2.

System Design

The scattering microscopy setup is based on an inverted microscope with a reflected darkfield illumination. Thus, only light that is scattered by the sample is collected in a certain angular range that depends on the objective and the effective numerical aperture defined by the aperture stop in $F^{\prime }$ (see Fig. 1). In the following, the angles $\vartheta$ and $\phi$ are defined relative to the $z$-axis, which equals the optical axis of the objective.²⁵ The positive $z$-direction is the direction of the detection path. The $x$- and $y$-directions are defined by the motorized microscopy stage (see the inset in the upper left corner in Fig. 1). So, the angle $\vartheta$ is the polar angle and $\phi$ is the azimuthal angle (in air) relative to the optical axis. As described by Schmitz et al.,²⁵ the unidirectional illumination is realized by a collimated beam with well-defined angles ${\vartheta }_I=121\mathrm{deg}$ and ${\phi }_I=341\mathrm{deg}$.

Fig. 1

Experimental setup for angular and spectrally resolved scattering microscopy. The scattered light is represented by solid orange lines. In the spectral mode path (S; dashed yellow lines), the scattered light is focused onto a fiber. In the angular mode path (A; dashed red lines), the back focal plane is focused onto a CCD camera.

In the angular mode, a supercontinuum laser source (SuperK Blue, NKT Photonics A/S, Birkerød, Denmark) in combination with an acousto-optic tunable filter (AOTF-PCAOM Vis, Crystal Technology, LLC, Palo Alto, California) illuminates the sample placed in the object plane $O$ quasi-monochromatically.²⁶ Ultrasonic waves generated by a piezoelectric transducer modulate the refractive index in the birefringent crystal of the AOTF. Hence, the incoming supercontinuum beam splits into its appropriate diffraction orders. The first diffraction order is coupled into a polarization-maintaining fiber (P1-630PM-FC-2, Thorlabs, Newton, New Jersey) and the collimated output beam of the fiber irradiates the sample. By tuning the ultrasonic frequency and power the optical wavelength can be adjusted continuously between 420 and 700 nm. In Fig. 1, the angular mode (A) is represented by dashed red lines. A part of the backscattered light by the sample is collected by the objective. In the angular mode, a long distance objective (LD EC Epiplan-Neofluar 50x/0.55 HD DIC M27 air, Carl Zeiss AG, Oberkochen, Germany) detects the scattered light in an angular range from $\mathrm{\Theta }=89\mathrm{deg}$ to $\mathrm{\Theta }=153\mathrm{deg}$ with an effective numerical aperture ${\mathrm{NA}}_{\mathrm{eff}}$ of 0.53 (${\vartheta }_{\max }=32\mathrm{deg}$). The scattering angle $\mathrm{\Theta }$ is the angle between the incident and the scattered light. The back focal plane $F$ of the objective represents the angular distribution of the collected scattered light. Via a lens system it is focused onto a CCD camera (SIS p1010, Theta System GmbH, Gröbenzell, Germany) placed in the Fourier plane $F^{\prime \prime }$. Thus, each pixel on the CCD chip indicates a certain scattering angle with a resolution of about $0.2\mathrm{deg}/\text{pixel}$ close to the optical axis. In the Fourier plane, the von Bieren condition describes the spatial frequency, i.e., the pixel position, to be proportional to $\sin (\vartheta )$,²⁷ where $\vartheta$ is the polar angle between the scattered light and the optical axis. The construction of the lens system is precisely described in Schmitz et al.²⁵ The additional Fourier lens $L_3$ ($f_3=60\mathrm{mm}$) is placed 100 mm behind the intermediate plane $F^{\prime }$. The Fourier plane $F^{\prime \prime }$ can be found 150 mm behind $L_3$.

For spectrally resolved measurements, the sample is illuminated by a broadband light source provided by the supercontinuum laser. The spectral mode (S) is illustrated in Fig. 1 with dashed yellow lines. An objective with a long working distance (Epiplan 5x/0.16, Carl Zeiss AG, Oberkochen, Germany) with a numerical aperture of $\mathrm{NA}=0.16$ (${\vartheta }_{\max }=9.2\mathrm{deg}$) detects the scattered light in an angular range from about $\mathrm{\Theta }=112\mathrm{deg}$ to $\mathrm{\Theta }=130\mathrm{deg}$. One end of a fiber with a core diameter of 1000 μm is placed in the image plane $O^{\prime \prime }$ and connected to a CCD spectrometer (MCS-CCD-Lab, Carl Zeiss AG, Oberkochen, Germany) recording the spectrum of the detected scattered light. By sliding a mirror into the path a RGB camera (NS1300CU, NET GmbH, Finning, Germany) visualizes the object in the image plane $O^{\prime \prime }$, allowing the positioning of the sample with respect to the center of the fiber. With an overall magnification of $12.5\times$ the fiber acts as a field stop with an aperture of 80 μm. A small NA is advantageous for spectrally resolved measurements since the integration over a small angular range does not average out the spectral oscillations and therefore the information content. For angular resolved measurements a high NA is favored to detect a large angular range. To alter between both, angular and spectral mode path, the AOTF is replaced and the relevant flip mirrors are toggled. Hence, without influencing the sample position, the measurement in both modes is offered smoothly.

3.

Theoretical Backscattering Model Based on Mie Theory

The theoretical calculations of the angular and spectral dependencies of light scattering in this study are based on Mie theory.^24,28 It provides an exact analytical solution to Maxwell’s equations describing the scattering of an electromagnetic wave by a homogeneous, spherical particle.²³ With known input parameters, Mie calculations enable the determination of the phase function $p(\mathrm{\Theta })$ and the scattering cross-section $C_s$. These input parameters are the sphere diameter $D$, the wavelength $\lambda$ in vacuo, and the refractive indices of the scatterer $n_S$ and of the surrounding medium $n_M$. The model described in Schmitz et al.²⁵ is valid for spectrally resolved measurements but only for air as medium surrounding the scatterer. Thus, the model was extended and generalized for calculations of light scattering by particles in an arbitrary medium with known dispersion.

The theoretical spectra and angular distributions are calculated by means of the Stokes vectors of the incident ${\overrightarrow{S}}_i$ and the scattered light ${\overrightarrow{S}}_s$, respectively. The Stokes-Mueller calculus for the scattering microscopy setup is

These dependencies on the refractive index show the impact of dispersion on the detectable scattering angle $\mathrm{\Theta }$. Thus, the range of the detectable scattering angle varies with the wavelength.

The rotation matrix $\mathbf{R}$ considers the rotation of the polarization state relative to the plane of scattering spanned by the vectors ${\overrightarrow{k}}_i$ and ${\overrightarrow{k}}_s$.^24,25 The angle between the plane of scattering and the plane of incidence $\xi =\xi ({\vartheta }_s,{\phi }_s)$ depends on the angles of the scattering direction in the medium. Hence, the rotation matrix $\mathbf{R}$ varies for each wavelength. The plane of incidence is spanned by the incident wave vector ${\overrightarrow{k}}_i$ and the $z$-axis.²⁵ The plane of scattering is rotated about the scattered wave vector. For ${\phi }_s={\phi }_i$ and ${\phi }_s={\phi }_i-180\mathrm{deg}$ both planes are identical.

As explained in Sec. 2, the setup is an inverted microscope with a reflected darkfield illumination where the samples are placed upon a coverslip. For that reason, the transmission through the coverslip of the illumination and the light scattered by the sample have to be considered.²⁵ Based on Fresnel’s formulas the transmission matrices $\mathbf{T}$ have to be calculated for each interface.²⁹ Four matrices $\mathbf{T}$ are required, one for each interface: for the illumination the transmission matrices from air to the coverslip (${\mathbf{T}}_{\mathrm{AG}}$, air to glass) and from the coverslip to the medium (${\mathbf{T}}_{\mathrm{GM}}$, glass to medium) and for the light scattering the transmission back from the medium to the coverslip (${\mathbf{T}}_{\mathrm{MG}}$) and from the coverslip to air (${\mathbf{T}}_{\mathrm{GA}}$) have to be examined. With air as the medium surrounding the scatterer both the matrices ${\mathbf{T}}_{\mathrm{AG}}$ and ${\mathbf{T}}_{\mathrm{GM}}$ and the matrices ${\mathbf{T}}_{\mathrm{MG}}$ and ${\mathbf{T}}_{\mathrm{GA}}$ are equal. The transmission matrices are also wavelength-dependent due to dispersion. Multiple reflections between both interfaces can be neglected due to low intensity.

The detectors for both angular and spectrally resolved measurements, i.e., the CCD camera and the CCD spectrometer, are not sensitive to polarization. Thus, to determine the intensity of the scattered light, only the first Stokes parameter $S_{s,0}$ of the scattered Stokes vector ${\overrightarrow{S}}_s$ is of importance. The incident angles ${\vartheta }_I$ and ${\phi }_I$ are kept constant for the entire experiment. In the angular mode, the incident light is linearly polarized with ${\overrightarrow{S}}_i={(\begin{array}{cccc}1& -1& 0& 0\end{array})}^{\mathrm{T}}$ since the AOTF’s output is linearly polarized. The theoretical angular distribution of the scattered light is estimated for the detectable angular range by applying a single wavelength $\lambda$ and the corresponding incident angles

The intensity $I_{T,\lambda }(u,v,D)$ for each pixel on the CCD chip with the Cartesian coordinates $(u,v)$ is obtained via the von Bieren condition, by transforming Eq. (5) with

The incident light for spectral measurements is unpolarized as the supercontinuum laser provides an unpolarized output. Thus, its Stokes vector is ${\overrightarrow{S}}_i={(\begin{array}{cccc}1& 0& 0& 0\end{array})}^{\mathrm{T}}$. The scattered light from all the collected angles is integrated and detected spectrally resolved. To take into account the spectral resolution of the spectrometer, the spectrum is convolved with a Gaussian function $g(\lambda )$ with a full width half maximum (FWHM) of $\mathrm{\Delta }\lambda =3.08\mathrm{nm}$, measured at the wavelength $\lambda =632.8\mathrm{nm}$. By applying ${\phi }_i$ and ${\vartheta }_i$ depending on the dispersion, the theoretical scattering spectrum $I_T(\lambda ,D)$ of a single sphere with diameter $D$ measured by the scattering microscope is

4.

Materials and Methods

5.

Results and Discussion

Scattering patterns were measured on several single polystyrene beads from the working suspensions PS4-2 and PS2-8 in angular and spectrally resolved mode, respectively. In the following, the single particles are numbered and denoted with the number sign “#.” The angular distribution of the scattered intensity at a particular wavelength, e.g., $\lambda =620\mathrm{nm}$, for one single particle PS4-2#1 is shown in Fig. 2. In the upper left of the figure [Fig. 2(a)] the measured distribution $I_{E,620}(u,v)$ is displayed as detected with the CCD camera. Utilizing the correlation algorithm (Sec. 4.2), the sphere diameter was identified to be $D_{620}=4.125\mu \text{m}$. The corresponding theoretical distribution $I_{T,620}(u,v,D_{620})$ is shown in Fig. 2(b). These two images represent the Fourier plane. In Fig. 2(c) and 2(d), the experimental pattern $I_{E,620}(\vartheta ,\phi )$ and the theoretical pattern $I_{T,620}(\vartheta ,\phi ,D_{620})$ are plotted versus the angles $\vartheta$ and $\phi$. The experimental data is normalized onto the theoretical data. Both experiment and theory are in very good agreement. The maximum of the correlation function $C_{620}(D)$ indicates the best fit, as shown in Fig. 2(e). Due to the broad peak of the correlation function, the accuracy of the diameter estimation is relatively low compared to spectrally resolved measurements. Therefore, in order to increase the accuracy, the diameter of the single particle was determined for each wavelength measured between 420 and 700 nm. The mean diameter $D_{\text{mean}}$ is calculated by averaging these 57 estimated diameters. For PS4-2#1, the mean value is $D_{\mathrm{mean}}=4.1200\mu \text{m}$ and the standard deviation $\sigma$ is 0.0094 μm. Consequently, by averaging the diameter values, a very high accuracy can be achieved.

Fig. 2

Angular distribution of the scattered intensity for sample PS4-2#1 at a particular wavelength ($\lambda =620\mathrm{nm}$). (a) Fourier plane measurement $I_{E,620}(u,v)$. (b) Best fit theory $I_{T,620}(u,v,D_{620})$. (c) Experimental $I_{E,620}(\vartheta ,\phi )$ and (d) theoretical $I_{T,620}(\vartheta ,\phi ,D_{620})$ scattering pattern versus scattering angles. (e) Correlation function $C_{620}(D)$ with the maximum at $D_{620}=4.125\mathit{\mu }\text{m}$.

To compare the two resolution modes of the scattering microscopy setup, the spectrum of the scattered light of exactly the same single particle was measured. The scatter spectrum $I_E(\lambda )$ of PS4-2#1 is represented by the light blue line in Fig. 3. Additionally, the best theoretical fit $I_E(\lambda ,D_{\mathrm{spec}})$ is shown as a dark blue line. Anew, the experimental curve is normalized onto the theoretical curve. The modulation of the characteristic Mie oscillations show some dissimilarity. Nonetheless, the spectral positions of the Mie oscillations of the experimental curve coincide with the theoretical curve very well. The diameter of PS4-2#1 determined from the spectrally resolved measurement is $D_{\mathrm{spec}}=4.128\mu \text{m}$. In the spectral mode, the accuracy of the diameter estimation is higher than in the angular mode. The main lobe of the correlation function in the spectral mode has a FWHM of less than 30 nm.²⁵ The value of $D_{\mathrm{spec}}$ differs slightly from the value of $D_{\text{mean}}$. However, the difference is less than the standard deviation $\sigma$ and the diameter values show a very good agreement.

Fig. 3

Scatter spectrum $I_E(\lambda )$ of a single bead PS4-2#1 (light blue line) and theoretical spectrum $I_E(\lambda ,D_{\mathrm{spec}})$ for a sphere diameter $D_{\mathrm{spec}}=4.128\mathit{\mu }\text{m}$ (dark blue line).

The results of the presented approach are reproducible for any single particle in the working suspension. In Fig. 4, the measured and theoretical scattering patterns for a single polystyrene bead PS2-8#5 are compared. On top of Fig. 4(a) and 4(b) the experimental $I_{E,530}(\vartheta ,\phi )$ and theoretical $I_{T,530}(\vartheta ,\phi ,D_{530})$ angular distributions for the wavelength $\lambda =530\mathrm{nm}$ are shown with a determined sphere diameter $D_{530}=2.742\mu \text{m}$. The mean diameter averaged over the wavelengths is $D_{\text{mean}}=2.7394\mu \text{m}$ with a standard deviation $\sigma =0.0063\mu \text{m}$. In the plot below [Fig. 4(c)], the measured spectrum of PS2-8#5 is compared against the theoretical spectrum for the best fit diameter $D_{\mathrm{spec}}=2.739\mu \text{m}$. In this case, the deviation between $D_{\text{mean}}$ and $D_{\mathrm{spec}}$ is merely 0.4 nm, confirming the accordance of angular and spectrally resolved measurements once more.

Fig. 4

Comparison of experiment and theory for sample PS2-8#5. (a) Experimental $I_{E,530}(\vartheta ,\phi )$ and (b) theoretical $I_{T,530}(\vartheta ,\phi ,D_{530})$ scattering pattern versus scattering angles with $D_{530}=2.742\mathit{\mu }\text{m}$. (c) Measured spectrum $I_E(\lambda )$ (light blue line) and theoretical spectrum $I_E(\lambda ,D_{\mathrm{spec}})$ with $D_{\mathrm{spec}}=2.739\mathit{\mu }\mathrm{m}$ (dark blue line).

A statistical analysis of the results for several single particles can be performed with the help of a box plot. In the following, 5 selected single particles for each type of polystyrene beads are analyzed exemplarily. The box plots for these particles are shown in Fig. 5. They show the distribution of the estimated diameters $D_{\lambda }$ from angular resolved measurements for each sample. The central marks in the box identify the median value of the diameters $D_{\text{median}}$, which is less sensitive to outliers. The median diameters $D_{\text{median}}$ differ slightly from the mean values $D_{\text{mean}}$. However, Table 1 demonstrates deviations between median and mean of magnitude of 1 nm or less. The boxes themselves are limited by the lower and upper quartiles, i.e., the interquartile range characterizing the middle 50% of the diameter values. Figure 5 shows the boxes for each sample, whereas the interquartile ranges of $\approx 10\mathrm{nm}$ are rather small, which confirms the quality of the approach. The whiskers indicate diameter values lying outside of the interquartile range not considering outliers that are plotted individually. In addition, the diameter values $D_{\mathrm{spec}}$ determined from spectrally resolved measurements are marked in Fig. 5 by the asterisks. These diameters $D_{\mathrm{spec}}$ are located within the boxes in the most cases. A summary of the estimated sizes of the 5 exemplary particles for angular and spectral mode is given in Table 1. As previously mentioned it shows a marginal deviation of 1 nm or less between mean and median diameters. In general, for each single bead measurement the standard deviations in the angular mode are less than 10 nm, the relative standard deviations are less than 0.25%. Therefore, in the angular mode a very high accuracy in the mean diameter estimation is obtained. Furthermore the deviation between the estimated values in angular and spectral mode is less than 0.20% for PS4-2 and less than 0.13% for PS2-8, respectively. The slightly greater deviations for the samples PS4-2 may be caused by the dissimilarities between the theoretical and experimental spectra which are larger for PS4-2 than for PS2-8. The response function of the spectrometer was supposed to be a Gaussian with a FWHM of 3.08 nm for the whole spectral range, though the FWHM might depend on the wavelength, which might have a larger impact on the spectra with more oscillations. However, the estimated diameter values in the angular and the spectral mode are in agreement within the measurement accuracy. In addition, the determined particle sizes are located in the range of size distributions validated with a collimated transmission setup.^25,33 The nominal values (see Sec. 4.1) given by the manufacturer appear to be overestimated.

Fig. 5

(a) Statistical analysis of the estimated diameter values $D_{\lambda }$ for 5 samples of PS4-2 and (b) 5 samples of PS2-8. Diameters determined from spectrally resolved measurements $D_{\mathrm{spec}}$ are plotted as red asterisks.

Table 1

Particle sizes determined in angular and spectral mode.

6.

Conclusion

For the evaluation of a novel scattering microscopy setup combining angular and spectral resolution mode, measurements in both modes were compared with each other. Based on elastic light scattering in the backward direction, the diameters of single polystyrene beads were distinguished. The experimental angular and spectral scattering patterns were analyzed by correlating them with the theoretical scattering patterns, which were computed by using Mie theory. In the angular mode, the diameter of a single bead was determined for multiple wavelengths and averaged subsequently. Mean diameters were calculated with a standard deviation of less than 10 nm. The relative standard deviations are 0.25% or less. By analyzing the spectrally resolved measurements, diameters of exactly the same beads were determined and compared with the diameters achieved by the angular resolved mode. The results of the investigations in angular and spectral resolution modes show an excellent agreement. The deviation between the two methods is only 0.20% or less. Thus, the combination of the angular and the spectral resolution mode results in a very high accuracy for particle characterization. In the literature other validations of angular resolved and spectroscopic scattering measurements show deviations that are 10 times larger, even though comparable suspensions were utilized.¹⁰ The differences may originate from the differences in the setup, i.e., mode and direction of the illumination.

In this study, the investigations in the two resolution modes have figured out that changes in the particle size have stronger influence on spectral patterns than on angular patterns. These changes result in a shift or a modification of the characteristic oscillations in the scatter spectrum. The advantage of the spectral mode can be quantified with the FWHM value of the correlation functions. In contrast, the advantage of the angular resolution mode is the higher sensitivity to changes in the particle shape, structure and orientation, which will be shown in a following contribution. Preliminary experiments with cylindrical samples, e.g., fiber glass, also show a good agreement in size prediction using both methods. Therefore, the presented system with the utilization of the advantages of both angular and spectral resolution mode creates new opportunities in light scattering analysis of not only different spheres³⁴ but also biological samples.^11,35,36 Light scattering by cells and the contribution of the intracellular organelles could be simulated using Mie theory models with Gaussian, exponential, or log normal distributions of scatterer sizes.^37,38 These models allow the interpretation of differences in angular and spectrally resolved scattering patterns due to the internal structure of cells. Since cells or cell nuclei as well as organelles are rarely exactly spherical, analysis algorithms using Mie theory would be inappropriate. The light scattering patterns of spheroidal¹⁵ and ellipsoidal³⁸ scatterers can be calculated using the T-matrix method. Scatterers of arbitrary shape could be simulated using the discrete dipole approximation.³⁹ The focus of future work lies on the investigations of single cells and their temporal changes, e.g., due to the induction of apoptosis in living cells.¹⁸ Further steps are the observation of cell spheroids⁴⁰ and the comparison of these measurements with goniometric measurements.